thq_02-key - Exam 165 MATH 2451 thq_02 T-C Use Course...

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Unformatted text preview: Exam : 165 MATH 2451 thq_02 T-C- Use Course : Multivariable Calculus Due Date : 2016-01-26 Instructor : McCary KEZ (Flrst Name) (Last Name) 2016-01-19 14 : 10 1. (10 points} Let. c, s E R be nonzero. (a) Find the real eigenvalues or Show there are none. Is ,4 (cc/mm MMLT up) A ROTAT'MM. FM) =r/(Utr -A 7!“ (JV/0 = (A-cf‘ 4453’ P (Q) > 0 NO [(8, EMEWA Ls (b) Find the real eigenvalues or Show there are none. rm= pm=o =9 ¢1=1W r; A HyPE/zeoch 20THwa 1:5,) Mfiw (A; mime» To an 2. (10 points} Evaluate. They all end-up as “nice” expressions. You’ll see all of these later! (a) Easy. det [(20809) —T sin(9}] z A(m(9))x+jfi (9)): = fl sin(9} r c0s(n‘9} ( (b) Medium. f EXPAND on) Mrs ooLuMA} c0s(n‘9} —r sin(n‘9} f} 44;.(9) J‘Lwflo) 095(9) vim“) (423(9) 7M0» det sin(9} r c0s(n‘9} f} = 0 o - 0' o D 1‘ /' may) j; WW) I} f} 1 =1 (e) Hard: figure it out somewhere else and then neatly copy it here. .. sin(¢'} sin( } ,0 sin(¢'v} c0s(n‘9} ,0 c0s(¢'v} sin(n‘9} 5111(0) c0s(n‘9} —,0 5111(0) sin(9} ,0 c0s(¢'} c0s(n‘9} EXPAND 0"] TI”; '20“) det n9 c0s(¢'} f} —,0 5111(0) ’ ,fMWMo) memre) I _ “meme? —,ow~(®M(9) ‘ “(PM [fwmmm ,mmmm ‘°'M['v]* (—rm(¢))0“"3 MWWKQ ,aMW)m(93 _M(¢)4§,.(e) NW) (NW) _ M(W‘M(°) 'MWlWW) : (fault?) w) (M (a) MOP)M(0] '(“WW) MW)M‘M(0) W(\p)MH(97 =1)“th ( (wonmwmm + (WWW/mm») — WM (v) ((mmw on)“ + (w mflwefl“) W L————-'-4 s - ,9“ msQO m mm) _ f,“ Mm (9mm?) = - ’* mm (MW + M<)(w( a“) r ( have, = 1’me U 1 ' (a) Plot the parallelogram determined by Mg 51 and Mg 52 for the given value of t. . mm.- . nan- . inssa:. i=0 i=1 i=2 i=4 3. (10 points) Consider the matrix M3 = [1 1. known as a horizontal shear. It is very important to notice that det Mg : 1 which implies that M3 is area-preserving (all of your parallelograms should have the same area, which is interesting because you horizontally shear the unit square out to t = 10'12 without changing its area). (b) Consider the matrix A = [6.5], where (i and I; are given in the plot. i. Determine the matrix R which will rigidly rotate the the indicated parallelogram such that the image of ti will sit on the positive 33-axis. That is, the columns of RA : [Rin] will determine the new, rotated parallelogram. ii. Determine the matrix S which will horizontally shear RA into a rectangle. That is, columns of S RA will determine a rectangle. G) & :sfl' 9v? go 131mm 8y ‘1}: = mi...‘ all J?» g a, -42 E "a? 0‘1 a 7 Procedures such as this are related to matrix factorizations which are varied and important. The purpose here is to make sure you understand the associated geometry of what is happening and not just the (important, but boring) steps of something like an L U factorization. 4. (10 points} The shaded regions are ellipses centered at the origin. {a} Find the matrix R which performs the following mapping, and compute det R. an (4.2) vhf-2) N we) we.» r; a it _ E a, (b) Find the matrix L which performs the following mapping, and compute detL. [a] {c} Using the previous parts, find the matrix M which performs the following mapping, and computedetM. m 6 N5 0? 9112, r. N E M 1‘5 or IN mm o @ M r: R L ~—> w [M] = 1.,[5 Elf °] W" 4 comm! EWW oA/ rm Wide: M = WU ' 3 N Ms 1:? 51/ _ 4 [Era 4] 4 1 so - ’9' 3 9.13 M =L‘ E' [M] = [R]"[L]" , . (fiin MOW wHkT THIS WOULD MM! = R '1 so . VTélM—LL/j l] 09])Re(5:(a31_ 1 -1 =[Rroa] New MO=H€ o , 4 {2 (d) 'Nhat is the area of the tilted ellipse? ' L O ( mm CIPLLE) DfirfflMINANT Is Me ewe mm. m 7% MM chum: = O (Dru/He) Am ® ‘77 Mlflllmfi’g = 9 THE “95”” ‘5 NOT 2* T’fiLTPDEmee, This page won’t. be graded. ...
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